About the strength of a perturbing potential

[ftft]

I often hear scientist say that generally the effect of a weak perturbing potential on the ground state of a system is larger than its effect on higher states and that the strength of the weak perturbation on the states decrease as we go higher in energy. For example, if a weak potential is removing degeneracy of states we will always find the strongest splitting happening in the lowest degenerate state. I can't find a reference to this assumption. Is it true anyway? And why so??

 

[hilbert2]

I often hear scientist say that generally the effect of a weak perturbing potential on the ground state of a system is larger than its effect on higher states and that the strength of the weak perturbation on the states decrease as we go higher in energy. For example, if a weak potential is removing degeneracy of states we will always find the strongest splitting happening in the lowest degenerate state. I can't find a reference to this assumption. Is it true anyway? And why so??

Well, for example, if you have a particle in a box and you add a perturbation that makes the bottom of the potential well slightly tilted instead of flat, you can see by classical analogy that the slight change in potential when going from one end of the box to another doesn't really affect a particle that has a large kinetic energy.